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What You Should Know about the Common Core Standards Mathematics Appendix

Geometry Concepts Your High School Student Should Know for Common Core Standards

To meet the Common Core Standards for mathematics, high school students must be familiar with some geometry. Geometry is a branch of mathematics that explores the nature and properties of points, lines, planes, and a host of shapes, including rectangles, triangles, circles, and spheres.

Congruence

Students show that two shapes are congruent, indicating that all sides and angles of the shape are exactly the same. Students also use rigid motions to move one shape on top of another to show that all parts are the same. Rigid motions involve moving an object without changing its size or shape.

Take a look at the example of rigid motion (reflection and translation in this instance) in the coordinate plane. Notice that the size of the shape hasn't changed even though it is reflected over to the left and translated down.

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Circles

Common Core Standards require that upon graduation students know circles inside and out. Expectations include the following:

  • Explain why all circles are similar.

  • Draw inscribed angles, radii, tangents, and chords and explain how they're related.

  • Know that a circle's radius is perpendicular to the tangent at the point where the radius touches the circle.

  • Find the area of any given sector of a circle (to envision a sector, imagine a slice cut out of a round pizza).

  • Find the length of an arc on the perimeter of a circle (the length of the curved side of a sector). (High-school students already know how to find the circumference — the length all the way around a circle.

Draw a circle with a radius of six inches, with a sector that has an interior angle of 45 degrees, and ask your child to calculate the area of the sector. You'll need to use a compass and protractor to be precise with your measurements.

The formula for the area of the entire circle is A = πr2 so A = π × 62 = 3.14159 × 32 = 100.53 square inches. A circle is 360 degrees, so the area of the sector is 45/360 times the total area of the circle. The fraction 45/360 reduces to 1/8, so the area of the sector is 100.53 × 1/8 = 12.57 rounded to the nearest hundredth.

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Using the same circle, ask your child to find the length of the sector's arc.

To find the arc, your child must calculate the circumference of the circle and then multiply it by the same fraction: 1/8. The formula for the circumference of a circle is C = πd where d is diameter. Diameter is twice the radius of 6 inches or 12 inches, so C = π × 12 = 37.70 rounded to the nearest hundredth.

So 1/8 of that is about 4.71.

Geometric measurement and dimension

Geometric measurement and dimension standards call on students to use formulas for the volumes of three-dimensional figures, including spheres, cones, cylinders, and pyramids:

  • Volume of a sphere: 4/3πr2 where r is the radius of the sphere

  • Volume of a cone: 1/3πr2h where r is the radius of the cone's base and h is its height

  • Volume of a cylinder: πr2h where r is the radius of the cylinder's base and h is its height

  • Volume of a pyramid: 1/3bh where b is the area of the pyramid's base and h is its height

Ask your child to calculate the volume of a cylinder that's 8 centimeters in height and 10 centimeters in diameter. To solve this problem, plug the numbers into the formula for the volume of a cylinder: πr2h = π × 102 × 8 = 3.14159 × 100 × 8 = 3.14159 × 800 = 2513.27 cubic centimeters.

Geometric modeling

Modeling with geometry calls on students to apply geometric concepts to real-world situations, such as the following:

  • Estimate the volume of an aboveground swimming pool using the formula for the volume of a cylinder.

  • Use formulas for area and volume to calculate population density in a given environment.

  • Design a structure using various geometric methods to achieve a specific goal, such as using the least amount of building materials.

Have your child calculate the volume or the space inside various three-dimensional objects or areas around your house. For example, you may have your child calculate the number of gallons of water your water heater holds (without peeking at the number on the water heater, of course).

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